Filling of closed surfaces

Let F-g denote a closed oriented surface of genus g. A set of simple closed curves is called a filling of F-g if its complement is a disjoint union of discs. The mapping class group Mod(F-g) of genus g acts on the set of fillings of F-g. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of F-g are in the same Mod(F-g)-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of F-2 whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of Mod(F-2). We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of F-2 is two. Finally, given positive integers g and k with (g, k) not equal (2, 1), we construct a filling pair of F-g such that the complement is a union of k topological discs